A compact complex manifold is called in Fujiki class $\mathcal C$ if it is bimeromorphic to a compact Kähler manifold, or equivalently, if there exists a proper holomorphic bimeromorphic map (i.e. a holomorphic modification) $\mu:\tilde X\to X$ such that $\tilde X$ is a compact Kähler manifold. Another characterization is that if and only if $X$ admits a Kähler current, that is a closed (1,1) current $T$ satisfying $T\ge\varepsilon\omega$ for some real number $\varepsilon>0$ and some positive Hermitian form $\omega$ (see for example Demailly-Paun 04, p.1263).

As we know the de Rham class $[T]$ of the Kähler current $T$ is also representable by a smooth form $\alpha$, such that $[\alpha]=[T]\in H^{1,1}(X,\mathbb R)$, then what property does $\alpha$ have? Of course it should not be positive, otherwise the manifold is already Kähler, but except that, what other properties does $\alpha$ have, can we always find a semi-positive $\alpha$ to represent the class $[T]$ of a Kähler current $T$?